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NASA NTRS1992

Polarimetric signatures of a canopy of dielectric cylinders based on first and second order vector radiative transfer theory

Complete polarimetric signatures of a canopy of dielectric cylinders overlying a homogeneous half space are studied with the first and second order solutions of the vector radiative transfer theory. The vector radiative transfer equations contain a general nondiagonal extinction matrix and a phase matrix. The energy conservation issue is addressed by calculating the elements of the extinction matrix and the elements of the phase matrix in a manner that is consistent with energy conservation. Two methods are used. In the first method, the surface fields and the internal fields of the dielectric cylinder are calculated by using the fields of an infinite cylinder. The phase matrix is calculated and the extinction matrix is calculated by summing the absorption and scattering to ensure energy conservation. In the second method, the method of moments is used to calculate the elements of the extinction and phase matrices. The Mueller matrix based on the first order and second order multiple scattering solutions of the vector radiative transfer equation are calculated. Results from the two methods are compared. The vector radiative transfer equations, combined with the solution based on method of moments, obey both energy conservation and reciprocity. The polarimetric signatures, copolarized and depolarized return, degree of polarization, and phase differences are studied as a function of the orientation, sizes, and dielectric properties of the cylinders. It is shown that second order scattering is generally important for vegetation canopy at C band and can be important at L band for some cases.

Tsang, Leung↗

NASA NTRS1987

Variational approach to probabilistic finite elements

Probabilistic finite element method (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties, and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗

NASA NTRS1991

Variational approach to probabilistic finite elements

Probabilistic finite element methods (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗

NASA NTRS1986

Numerical advection by conservation of second-order moments

A new, accurate, and nondiffusive method for three-dimensional advection of trace species is presented. The method preserves tracer structures by conserving the second-order moments of the spatial distribution of tracer during advection. Upstream transport and second-order tracer distribution are described, and the moments of the tracer distribution about the center of a grid box are formally defined and related to the polynomial distribution. Formulas are presented which describe how the moments of a grid box are decomposed into a unique set of moments centered about each subbox and how they are reassembled into new grid boxes. A one-dimensional example if tracer transport is given, and limits necessary to maintain positive tracer concentrations are derived. The accuracy and stability of this method are analytically examined, and numerical experiments testing the effective resolution are presented. The method is compared with other methods for numerical advection of tracers.

Prather, M. J.↗

NASA NTRS1988

Kuhn-Tucker optimization based reliability analysis for probabilistic finite elements

The fusion of probability finite element method (PFEM) and reliability analysis for fracture mechanics is considered. Reliability analysis with specific application to fracture mechanics is presented, and computational procedures are discussed. Explicit expressions for the optimization procedure with regard to fracture mechanics are given. The results show the PFEM is a very powerful tool in determining the second-moment statistics. The method can determine the probability of failure or fracture subject to randomness in load, material properties and crack length, orientation, and location.

Liu, W. K.↗

NASA NTRS2002

Effect of Random Geometric Uncertainty on the Computational Design of a 3-D Flexible Wing

The effect of geometric uncertainty due to statistically independent, random, normally distributed shape parameters is demonstrated in the computational design of a 3-D flexible wing. A first-order second-moment statistical approximation method is used to propagate the assumed input uncertainty through coupled Euler CFD aerodynamic / finite element structural codes for both analysis and sensitivity analysis. First-order sensitivity derivatives obtained by automatic differentiation are used in the input uncertainty propagation. These propagated uncertainties are then used to perform a robust design of a simple 3-D flexible wing at supercritical flow conditions. The effect of the random input uncertainties is shown by comparison with conventional deterministic design results. Sample results are shown for wing planform, airfoil section, and structural sizing variables.

Gumbert, C. R.↗

NASA NTRS2011

Coupling Between Waveguide-Fed Slot Arrays

Coupling between two waveguide-fed planar slot arrays has been investigated using full-wave analysis. The analysis employs the method-of-moments solution to the pertinent coupled integral equations for the aperture electric field of all slots. In order to compute coupling between two arrays, the input port of the first array is excited with a TE(sub 10) mode wave while the second one is match-terminated. After solving the moment method matrix equations, the aperture fields of all slots are obtained and thereby the TE(sub 10) mode wave received at the input port of the second array is determined. Coupling between two arrays is the ratio of the wave amplitude arriving in the second array port to the incident wave amplitude at the first array port. The coupling mechanism has been studied as a function of spacing between arrays in different directions, e.g. the electric field plane, the magnetic field plane, and the diagonal plane. Computed coupling values are presented for different array geometries. This work is novel since it provides a good understanding of coupling between waveguide-fed slot arrays as a function of spacing and orientation for different aperture distributions and array architectures. This serves as a useful tool for antenna design engineers and system engineers.

Rengarajan, Sembiam↗

NASA NTRS2005

Application of an Extended Parabolic Equation to the Calculation of the Mean Field and the Transverse and Longitudinal Mutual Coherence Functions Within Atmospheric Turbulence

Solutions are derived for the generalized mutual coherence function (MCF), i.e., the second order moment, of a random wave field propagating through a random medium within the context of the extended parabolic equation. Here, "generalized" connotes the consideration of both the transverse as well as the longitudinal second order moments (with respect to the direction of propagation). Such solutions will afford a comparison between the results of the parabolic equation within the pararaxial approximation and those of the wide-angle extended theory. To this end, a statistical operator method is developed which gives a general equation for an arbitrary spatial statistical moment of the wave field. The generality of the operator method allows one to obtain an expression for the second order field moment in the direction longitudinal to the direction of propagation. Analytical solutions to these equations are derived for the Kolmogorov and Tatarskii spectra of atmospheric permittivity fluctuations within the Markov approximation.

Manning, Robert M.↗

NASA NTRS1979

Alternative signal processing techniques for a pulsed laser Doppler velocimeter system

The laser Doppler velocimeter system (LDV) being tested at NASA MSFC, Alabama, relies on a bank of classical filters for processing the returned lidar signal. Due to the complexity of this conventional filter system, alternative signal processing methods have been proposed for obtaining the Doppler spectrum. The fast Fourier transform (FFT) and the pulse-pair estimator are two techniques which appear to offer some advantages over the existing system. The work reported employs digital computer simulation to compare and evaluate these two processing methods for determining the first and second moments of the LDV velocity spectrum.

Calhoun, M. D.↗

NASA NTRS1993

Optimal thresholds for the estimation of area rain-rate moments by the threshold method

Optimization of the threshold method, achieved by determination of the threshold that maximizes the correlation between an area-average rain-rate moment and the area coverage of rain rates exceeding the threshold, is demonstrated empirically and theoretically. Empirical results for a sequence of GATE radar snapshots show optimal thresholds of 5 and 27 mm/h for the first and second moments, respectively. Theoretical optimization of the threshold method by the maximum-likelihood approach of Kedem and Pavlopoulos (1991) predicts optimal thresholds near 5 and 26 mm/h for lognormally distributed rain rates with GATE-like parameters. The agreement between theory and observations suggests that the optimal threshold can be understood as arising due to sampling variations, from snapshot to snapshot, of a parent rain-rate distribution. Optimal thresholds for gamma and inverse Gaussian distributions are also derived and compared.

Short, David A.↗

NASA NTRS2004

An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical Solutions of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation

The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.

Manning, Robert M.↗

NASA NTRS1983

A procedure for combining acoustically induced and mechanically induced loads (first passage failure design criterion)

The combined load statistics are developed by taking the acoustically induced load to be a random population, assumed to be stationary. Each element of this ensemble of acoustically induced loads is assumed to have the same power spectral density (PSD), obtained previously from a random response analysis employing the given acoustic field in the STS cargo bay as a stationary random excitation. The mechanically induced load is treated as either (1) a known deterministic transient, or (2) a nonstationary random variable of known first and second statistical moments which vary with time. A method is then shown for determining the probability that the combined load would, at any time, have a value equal to or less than a certain level. Having obtained a statistical representation of how the acoustic and mechanical loads are expected to combine, an analytical approximation for defining design levels for these loads is presented using the First Passage failure criterion.

Crowe, D. R.↗

NASA NTRS1992

Stochastic stability properties of jump linear systems

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented.

Feng, Xiangbo↗

NASA NTRS2004

An Integrated Fuselage-Sting Balance for a Sonic-Boom Wind-Tunnel Model

Measured and predicted pressure signatures from a lifting wind-tunnel model can be compared when the lift on the model is accurately known. The model's lift can be set by bending the support sting to a desired angle of attack. This method is simple in practice, but difficult to accurately apply. A second method is to build a normal force/pitching moment balance into the aft end of the sting, and use an angle-of-attack mechanism to set model attitude. In this report, a method for designing a sting/balance into the aft fuselage/sting of a sonic-boom model is described. A computer code is given, and a sample sting design is outlined to demonstrate the method.

Mack, Robert J.↗

NASA NTRS1992

A numerical assessment of rough surface scattering theories. I - Horizontal polarization. II - Vertical polarization

A numerical evaluation is presented of the regime of validity for various rough surface scattering theories against numerical results obtained by employing the method of moments. The contribution of each theory is considered up to second order in the perturbation expansion for the surface current. Considering both vertical and horizontal polarizations, the unified perturbation method provides best results among all theories weighed.

Rodriguez, Ernesto↗

NASA NTRS2013

High-Order Space-Time Methods for Conservation Laws

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

Huynh, H. T.↗

NASA NTRS1979

Method for decomposing observed line shapes resulting from multiple causes - Application to plasma charge-exchange-neutral spectra

A method is given for decomposing the widths of observed spectral lines resulting from unresolved line splitting, additive kinetic processes of different types, instrumental broadening (slit function), Doppler broadening, etc. all superimposed. The second moments are used as measures of the various widths involved. The method is not applicable if dispersion type (Lorentz) broadening occurs. Application is made to plasma charge-exchange-neutral spectra of hydrogen, deuterium, and helium.

Patch, R. W.↗

DOE OSTI2025

Beyond the two-point correlation: Constraining primordial non-Gaussianity with density-perturbation moments

Constraining primordial non-Gaussianity (PNG) on the large-scale cosmic structure (LSS) is an important step in understanding properties of the early Universe, specifically in distinguishing between different inflationary models. Measuring PNG relies on evaluating the scale-dependent correlations in the density field. New summary statistics beyond the two- and three-point correlation functions in configuration space and their Fourier-space counterparts, the power- and bispectrum may provide increased sensitivity. We introduce a new method for extracting the PNG signal imprinted on the LSS by using the first three Gaussian moments of the normalized correlation in density perturbations, evaluated on varying distance scales. We aim to assess this method’s sensitivity to local PNG, parameterized by f NL . We performed spherical convolutions on a range of scales on dark-matter-halo simulations to measure the scale-dependent correlations in the density field. From these, we computed the first three moments and compared them to a model expectation vector, parameterized to the second power in f NL . Our method provides about 21% improvement in sensitivity to f NL with respect to using the two-point correlation function alone. Notably, we find that the second moment alone carries nearly as much constraining power as the mean, highlighting the potential of higher order statistics. Given its simplicity and efficiency, this framework is well suited for application to current and upcoming large-scale surveys such as the Dark Energy Spectroscopic Instrument (DESI).

early universe↗